AMC12 2011 A

AMC12 2011 A · Q6

AMC12 2011 A · Q6. It mainly tests Systems of equations, Arithmetic misc.

The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was $61$ points. How many free throws did they make?

一个篮球队的球员投中了一些三分球、两分球和一分的罚球。他们用两分球得到的得分与三分球相同。他们成功的罚球数比成功的两分球数多1。全队总得分为$61$分。他们投中了多少个罚球？

(A) 13 13

(B) 14 14

(D) 16 16

(E) 17 17

Answer

Correct choice: (A)

正确答案：（A）

Solution

For the points made from two-point shots and from three-point shots to be equal, the numbers of made shots are in a $3:2$ ratio. Therefore, assume they made $3x$ and $2x$ two- and three- point shots, respectively, and thus $3x+1$ free throws. The total number of points is \[2 \times (3x) + 3 \times (2x) + 1 \times (3x+1) = 15x+1\] Set that equal to $61$, we get $x = 4$, and therefore the number of free throws they made $3 \times 4 + 1 = 13 \Rightarrow \boxed{A}$

要使两分球得分与三分球得分相等，则命中球数之比为$3:2$。因此设他们分别投中$3x$个两分球和$2x$个三分球，从而罚球为$3x+1$个。总得分为\[2 \times (3x) + 3 \times (2x) + 1 \times (3x+1) = 15x+1\] 令其等于$61$，得$x = 4$，因此罚球数为$3 \times 4 + 1 = 13 \Rightarrow \boxed{A}$

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